## PIM Calculation.

To confirm that for the reduction of PIM there is no advantage in adding a resistor in parallel with the compensation capacitor to increase open-loop bandwidth I did a simplified calculation. I was only checking the effect of adding the resistor, and there are some nonlinear effects excluded by this simplification, so it is not 100% accurate.

Here is the circuit I started from, showing the simplification. The input stage is split into two separate stages so that the phase modulation can more easily be separated from the amplitude modulation. Vx is an amplitude modulated version of Vi, but this effect is what we want to ignore, and the phase difference between Vo and Vx turns out to be more useful because the modulation is then confined to the real part of the transfer function. The input stage is represented by a unity gain stage with a resistor from its output to the inverting driver stage, which is assumed to have sufficient open-loop gain so that its closed-loop gain is determined entirely by R and C. A large low frequency signal modulates the gain of the input stage so that its gain varies from 1-d to 1+d, where d is assumed to be small. A low level 20kHz signal then has its phase modulated by this gain variation.

Adding currents at the input of the driver stage,
Vx + BVo (1 +/-d) + Vo/R + Vo.jwC = 0
Therefore ... Vo ( B +/-Bd + 1/R + jwC ) = -Vx

The magnitude of the closed-loop phase shift is the phase angle of the expression in brackets. If the jwC part is assumed to be constant for now, then the effect of d on the real part is what we are interested in. Putting in some figures, suppose B = 0.05, to give a closed-loop amplifier gain of 20, and R = 200 gives 20dB negative feedback loop gain. If R has been chosen to give a 20kHz open-loop frequency response then wC = 1/200 = 0.005.

The real part now becomes ( 0.055 +/-0.05d ), a variation of +/- 0.9% if d = 0.01
With R omitted it becomes ( 0.050 +/-0.05d ), a variation of +/- 1% if d = 0.01
Assuming d is small, the change in phase has been reduced by about 10% by including R, so it does have some benefit. If the phase modulation was 0.11 deg without R it would be reduced to 0.10 deg with R.
Unfortunately the value of d is also affected by R, and to take an extreme example, with the low frequency component at 20Hz removing R will reduce the input stage signal by 1,000 times, and with distortion proportional to signal amplitude squared for a typical differential input stage d will fall by a factor of 1,000,000. Adding R therefore has two effects, one reduces phase modulation by a factor of 1.1, the other increases it by a factor of 1,000,000. Evidently R is a bad idea and should be omitted. The difference between the two effects may not be so extreme using different signal frequencies or different component values, and the calculation is not exact, but even so there seems little possibility of R reducing phase intermodulation effects.

A similar conclusion was reached in my article about TID, where including R raised the steady state error voltage above the transient error voltage and increased both transient and steady state distortion in the input stage.